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Properties

Label 2-34e2-17.9-c1-0-19
Degree $2$
Conductor $1156$
Sign $-0.275 + 0.961i$
Analytic cond. $9.23070$
Root an. cond. $3.03820$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.676 + 0.280i)3-s + (1.32 − 3.20i)5-s + (0.280 + 0.676i)7-s + (−1.74 − 1.74i)9-s + (−4.37 + 1.81i)11-s − 1.46i·13-s + (1.79 − 1.79i)15-s + (3.86 − 3.86i)19-s + 0.535i·21-s + (4.37 − 1.81i)23-s + (−4.94 − 4.94i)25-s + (−1.53 − 3.69i)27-s + (1.32 − 3.20i)29-s + (−5.72 − 2.37i)31-s − 3.46·33-s + ⋯
L(s)  = 1  + (0.390 + 0.161i)3-s + (0.592 − 1.43i)5-s + (0.105 + 0.255i)7-s + (−0.580 − 0.580i)9-s + (−1.31 + 0.546i)11-s − 0.406i·13-s + (0.462 − 0.462i)15-s + (0.886 − 0.886i)19-s + 0.116i·21-s + (0.911 − 0.377i)23-s + (−0.989 − 0.989i)25-s + (−0.294 − 0.711i)27-s + (0.246 − 0.594i)29-s + (−1.02 − 0.425i)31-s − 0.603·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.275 + 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.275 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1156\)    =    \(2^{2} \cdot 17^{2}\)
Sign: $-0.275 + 0.961i$
Analytic conductor: \(9.23070\)
Root analytic conductor: \(3.03820\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1156} (757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1156,\ (\ :1/2),\ -0.275 + 0.961i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.552006463\)
\(L(\frac12)\) \(\approx\) \(1.552006463\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 \)
good3 \( 1 + (-0.676 - 0.280i)T + (2.12 + 2.12i)T^{2} \)
5 \( 1 + (-1.32 + 3.20i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + (-0.280 - 0.676i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (4.37 - 1.81i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + 1.46iT - 13T^{2} \)
19 \( 1 + (-3.86 + 3.86i)T - 19iT^{2} \)
23 \( 1 + (-4.37 + 1.81i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (-1.32 + 3.20i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + (5.72 + 2.37i)T + (21.9 + 21.9i)T^{2} \)
37 \( 1 + (10.5 + 4.38i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (2.29 + 5.54i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (-8.76 - 8.76i)T + 43iT^{2} \)
47 \( 1 - 6.92iT - 47T^{2} \)
53 \( 1 + (0.656 - 0.656i)T - 53iT^{2} \)
59 \( 1 + (6.69 + 6.69i)T + 59iT^{2} \)
61 \( 1 + (-2.85 - 6.89i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + 1.07T + 67T^{2} \)
71 \( 1 + (2.02 + 0.840i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (-0.765 + 1.84i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (1.66 - 0.690i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (-6.69 + 6.69i)T - 83iT^{2} \)
89 \( 1 + 9.46iT - 89T^{2} \)
97 \( 1 + (3.41 - 8.24i)T + (-68.5 - 68.5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.214599605353705209776549672573, −9.008425166806110335184597092153, −8.078646350298396871417752106839, −7.25290325264930211675683651513, −5.84131242469744154993064627964, −5.27877785187410774627692358717, −4.55054788421398675815075590861, −3.14064074045583148888326586653, −2.14947643926354139004776962426, −0.60935028652092943295597539169, 1.85338561561460647775937631935, 2.89165020422722743005005287679, 3.43580425308207998314122047496, 5.19518847943290540026143914810, 5.72260923823222201682328123736, 6.93273807060848877981025925003, 7.44366683984468689570205352387, 8.315754181870875271774040354551, 9.213473178678705938078345683067, 10.32220166253127270955223204113

Graph of the $Z$-function along the critical line